Matrix convexity: Operator analogues of the bipolar and Hahn-Banach theorems
Let \(V\) and \(V'\) be dual vector spaces (over \(\mathbb{C}\)). This means that there is a non-degenerate pairing between the two spaces so that each of them can be seen as a subspace of the dual space of the other. Associated with the pairing come weak topologies on these vector spaces. If \(K\) is a convex subset of \(V\), the polar set \(K^0\) in \(V'\) is defined as \[ K^0= \{f\in V'\mid\text{Re}\langle v, f\rangle\leq 1\text{ for all }v\in K\}. \] It is a weakly closed convex set in \(V'\), containing \(0\). The classical bipolar theorem says that \(K^{00}= K\) if \(0\in K\) and \(K\) is weakly closed to begin with. The result is closely related to, and usually proved by means of the algebraic form of the Hahn-Banach theorem. This paper is a contribution to the further development of `quantized functional analysis'. It treats the quantized forms of the bipolar theorem and the Hahn-Banach theorem. The basic notion in this context is a matrix convex set, already introduced by \textit{G. Wittstock} in [North Holland Math. Stud. 90, 175-188 (1984; Zbl 0589.46003)]. As before, let \(V\) be a vector space (over \(\mathbb{C}\)) and consider the vector spaces \(M_n(V)\) of \(n\) by \(n\) matrices with entries in \(V\) for every \(n= 1,2,\dots\). A matrix convex set is a collection \((K_n)_{n= 1}\) of convex sets \(K_n\subseteq M_n(V)\) satisfying the following two conditions: i) If \(n\), \(m\) are natural numbers and \(\alpha\in M_{n, m}(\mathbb{C})\) such that \(\alpha^*\alpha= 1\), then \(\alpha^*K_n\alpha\subseteq K_m\). ii) For any \(n\), \(m\) we have \(K_n\oplus K_m\subseteq K_{n+ m}\). The quantized bipolar theorem now has the following form. If \(V\) and \(V'\) are dual vector spaces, there is a natural bilinear map from \(M_n(V)\times M_n(V')\to M_{nm}(\mathbb{C})\) given, and denoted by \[ \langle\langle v, w\rangle\rangle= [\langle v_{i,j}, w_{k,\ell}\rangle]. \] The matrix polar \(K^\pi\) of a matrix convex set \(K\) is now defined by \[ K^\pi_n= \{w\in M_n(V')\mid\text{Re}\langle \langle v,w\rangle\rangle\leq I_{nm}\text{ for any }v\in K_m\text{ and all }m\}. \] Here, the real part of a matrix \(x\) is \({1\over 2}(x^*+ x)\), where \(x^*\) is the adjoint matrix. Now \((K^\pi_n)_n\) is again a matrix convex set. Every \(K^\pi_n\) is weakly closed for the weak topology induced by the pairing \(\langle\langle\;,\;\rangle\rangle\) and \(0\in K^\pi_1\). Now, the bipolar theorem says that \(K^{\pi\pi}= K\) if and only if \(0\in K_1\) and \(K\) is weakly closed. The quantized form of the Hahn-Banach theorem is more involved. The theorem proved in the paper is rather general. Earlier versions, like the Hahn-Banach type extension theorems for completely bounded maps follow.
algebraic form of the Hahn-Banach theorem
quantized form of the Hahn-Banach theorem
Hahn-Banach type extension theorems for completely bounded maps
scientific article; zbMATH DE number 987559